Quantum Gravity Quantum Gravity is reputed to be one of the most difficult
puzzles of science. In practical terms it is probably of no direct relevance and
may even be impossible to verify by experiment. But for physicists it is the
holy grail which may enable them to complete the unification of all fundamental
laws of physics.

The problem is to put together general relativity and quantum mechanics into
one self consistent theory. The difficulty is that the two parts seem to be
incompatible, both in concept and in practice. Conceptually, it is the nature of
space and time which present fundamental differences. A direct approach,
attempting to combine general relativity and quantum mechanics, while ignoring
conceptual differences, leads to a meaningless quantum field theory with
unmanageable divergences.

There have, in fact, been many attempts to create a theory of quantum
gravity. In this article I will first outline the nature of general relativity
and quantum mechanics with emphasis on their similarities and differences. Then
I will briefly review some of the main stream approaches to quantum gravity.
Finally I will talk about some ways in which these ideas now seem to be
converging.

General Relativity General relativity is Einstein's monumental theory of
gravity. It is based on two fundamental principles: The principle of relativity
which states that all basic laws of physics should take a form which is
independent of any reference frame, and The principle of equivalence which
states that it is impossible to distinguish the effects of gravity from the
effects of being in an accelerated frame of reference.

Einstein struggled with the consequences of these principles for several
years, constructing many thought experiments to try to understand what they
meant. Finally he learnt about Riemann's mathematics of curved geometry and
realised that a new theory could be constructed in which the force of gravity
was a consequence of the curvature of space-time.

In constructing that theory, Einstein was not significantly influenced by any
experimental result which was at odds with the Newtonian theory of gravity. He
knew, however, that Newtonian gravity was inconsistent with his theory of
special relativity and he knew there must be a more complete self consistent
theory. A similar inconsistency now exists between quantum mechanics and general
relativity and, even though no experimental result is known to violate either
theory, physicists now seek a more complete theory.

In the decades that have followed Einstein's discovery, a number of
experimental confirmations of general relativity have been found but there still
remains a possibility that it may not be accurate on very large scales, or under
very strong gravitational forces. In any case, it is sure to break down under
the conditions which are believed to have existed at the big bang where quantum
gravity effects were important.

One of the most spectacular predictions of general relativity is that a dying
star of sufficient mass will collapse under its gravitational weight into an
object so compressed that not even light can escape its pull. These objects are
known as black holes. Astronomers now have a growing list of celestial objects
which they believe are black holes because of their apparent high density. The
accuracy of Einstein's theory may be stringently tested in the near future when
gravitational wave observatories such as LIGO come on-line to observe such
catastrophic events as the collisions between black holes.

Quantum Mechanics The Quantum theory was founded before Einstein began his
theory of relativity and took much longer to be completed and understood. It was
Planck's observations of quanta in the spectrum of black body radiation which
first produced signs that the classical theories of mechanics were due for major
revisions.

Unlike general relativity which was essentially the work of one man, the
quantum theory required major contributions from Bohr, Einstein, Heisenberg,
Schroedinger, Dirac and many others, before a complete theory of quantum
electrodynamics was formulated. In practical terms, the consequences of the
theory are more far reaching than those of general relativity. Applications such
as transistors and lasers are now an integral part of our lives and, in
addition, the quantum theory allowed us to understand chemical reactions and
many other phenomena.

In the 1960's and 70's, further discoveries in quantum field theory have led
to successful theories of the nuclear reactions and, in consequence, almost all
ordinary physical phenomena can now be attributed to quantum interactions, even
if the exact mechanisms are not always fully understood. The electromagnetic and
weak nuclear interactions are unified into one force while the strong nuclear
interaction is a force of a similar nature known as a gauge theory. Together
these forces and all observed particles are combined into one self consistent
theory known as the standard model of particle physics.

Despite such spectacular success, confirmed in ever more detail in high
energy accelerator experiments, the quantum theory is still criticised by some
physicists who feel that its indeterministic nature and its dependency on the
role of observer suggest an incompleteness.

Unification Since Newton set the foundations of physics, progress has come
mostly in the form of unification. Maxwell unified electricity, magnetism and
light into one theory of electromagnetism. Einstein unified space, time and
gravity into one theory of general relativity. More recently, the nuclear forces
have been (partially) unified with the electromagnetic force by Weinberg and
others.

According to conventional wisdom among physicists, the process of unification
will continue until all physics is unified into one neat and tidy theory. There
is no a priori reason to be so sure of this. It is quite possible that
physicists will always be discovering new forces, or finding new layers of
structure in particles, without ever arriving at a final theory. It is quite
simply the nature of the laws of physics as we currently know them that inspires
the belief that we are getting closer to the end.

After physicists discovered the atom, they went on to discover that it was
composed of electrons and a nucleus, then that the nucleus was composed of
protons and neutrons, then that the protons and neutrons were composed of
quarks. Should we expect to discover that quarks and electrons are made of
smaller particles? This is possible but there are a couple of reasons to suppose
not. Firstly there are far fewer particles at this level than there ever were at
higher levels. Secondly, their interactions are described by a clean set of
gauge bosons through renormalisable field theories. Composite interactions, such
as pion exchange, do not take such a tidy form. These reasons in themselves are
not quite enough to rule out the possibility that quarks, electrons and gauge
bosons are composite but they reduce the number of ways such a theory could be
constructed. In fact all viable theories of this type which have been proposed
are now all but ruled out by experiment. There may be a further layer of
structure but it is likely to be different. It is more common now for theorists
to look for ways that different elementary particles can be seen as different
states of the same type of object. The most popular candidate for the ultimate
theory of this type is superstring theory, in which all particles are just
different vibration modes of very small loops of string.

Note added: Just a few weeks after writing this, experimenters at Fermilab
announced the discovery of evidence for structure within quarks!

Physicists construct particle accelerators which are sort of like giant
microscopes. The higher the energy they can produce, the smaller the wavelength
of the colliding particles and the smaller the distance scale they probe. In
this way physicists can see the quarks inside protons, not through direct
pictures but through scattering data. Other things that happen as the energy
increases is that new heavy particles are formed and forces become unified. It
is impossible to be sure about what will happen the next time a new, more
powerful accelerator is built, but physicists can make theories about it.

In the next decade new accelerator experiments at CERN will probe beyond the
electro-weak scale. There is some optimism that new physics will be found but
nothing is certain.

Planck Scale At first sight it might seem ridiculous to suppose that we can
invent valid theories about physics at high energies before doing experiments.
However, theorists have already demonstrated a remarkable facility for doing
just that. The standard model of particle physics was devised in the 1960's and
experimentalists have spent the last three decades verifying it. The reason for
this success is that physicists recognised the importance of certain types of
symmetry and self-consistency conditions in quantum field theory which led to an
almost unique model for physics up to the electro-weak unification energy scale,
with only a few parameters such as particle masses to be determined.

The situation now is a little different. Experimentalists are about to enter
a new scale of energies and theorists do not have a single unique theory about
what can be expected. They do have some ideas, in particular it is hoped that
supersymmetry may be observed, but we will have to wait and see.

Despite these unknowns there are other more general arguments which tell us
things about what to expect at higher energies. When Planck initiated the
quantum theory he recognised the significance of fundamental constants in
physics, especially the speed of light (known as c) and his newly discovered
Planck constant (known as h). Scientists and engineers have invented a number of
systems of units for measuring lengths, masses and time, but they are entirely
arbitrary and must be agreed by international convention. Planck realised that
there should be a natural set of units in which the laws of physics take a
simpler form. The most fundamental constants, such as c and h would simply be
one unit in that system.

If one other suitable fundamental constant could be selected, then the units
for measuring mass, length and time would be determined. Planck decided that
Newton's gravitational constant (known as G) would be a good choice. Actually
there were not many other constants, such as particle masses known at that time,
otherwise his choice might have been more difficult. By combining c, h and G
Planck defined a system of units now known as the Planck scale. He calculated
that the Planck unit of length is very small, about 10-35 (ten to the power of
minus 35) metres. To build an accelerator which could see down to such lengths
would require energies about 1015 times larger than those currently available.
Note that units of speed and energy can be built from the three basic Planck
units but to measure temperature and charge as well we have to also set
Boltzmann's constant and the charge on the electron to one unit. In this way we
can devise a fundamental system of measurement for all physical quantities.

Physicists have since sought to understand what the Planck scale of units
signifies. Those who work with particles believe that at the Planck scale all
the four forces of nature, including gravity, are unified. Physicists who
specialise in general relativity have a different idea. In 1955 John Wheeler
argued that when you combine general relativity and quantum mechanics you will
have a theory in which the geometry of space-time is subject to quantum
fluctuations, He computed that these fluctuations would become significant if
you could look at space-time on length scales as small as the Planck length.
Sometimes physicists talk about a space-time foam at this scale but we don't yet
know what it really means. For that we will need the theory of quantum gravity.

Without really knowing too much for certain physicists guess that at the
Planck scale all forces of nature are unified and quantum gravity is
significant. It is at the Planck scale that they expect to find the final and
completely unified theory of the fundamental laws of physics.

The Small Scale Structure of Space-Time It seems clear that to understand
quantum gravity we must understand the structure of space-time at the Planck
length scale. In the theory of general relativity space-time is described as a
smooth continuous manifold but we cannot be sure that this is correct for very
small lengths and times. We could compare general relativity with the equations
of fluid dynamics for water. They describe a continuous fluid with smooth flows
in a way which agrees very well with experiment. Yet we know that at atomic
scales water is something very different and must be understood in terms of
forces between molecules whose nature is completely hidden in the ordinary
world. If space-time also has a complicated structure at the tiny Planck length,
way beyond the reach of any conceivable accelerator, can we possibly hope to
discover what it is?

If you asked a bunch of mathematicians to look for theories which could
explain the fluid dynamics of water, without them knowing anything about other
physics and chemistry, then they would probably succeed in devising a whole host
of mathematical models which work. All those models would probably be very
different, limited only by the imagination of the mathematicians. None of them
would correspond to the correct description of water molecules and their
interactions. The same might be true of quantum gravity. Nevertheless, the task
of putting together general relativity and quantum mechanics together into one
self consistent theory has not produced a whole host of different and
incompatible theories. The clever ideas which have been developed have enigmatic
things in common. It is quite possible that all the ideas are partially correct
and are aspects of one underlying theory which is within our grasp. It is time
now to look at some of those ideas.

Attempts to do Quantum Gravity The most direct way to try to quantise quantum
gravity is to use perturbative quantum field theory. This is a procedure which
has been applied with great success to electrodynamics. To do the same thing for
gravity it is necessary to first construct a system of non-interacting gravitons
which represent a zero order approximation to quantised gravitational waves in
flat space-time. These hypothetical gravitons must be spin two massless
particles because of the form of the metric field in general relativity.

The next step is to describe the interactions of these gravitons using the
perturbation theory of quantum mechanics, which are defined by a set of Feynman
diagrams derived from Einstein's gravitational field equations. For
electrodynamics this can be made to work, but only after conveniently cancelling
divergent anomalies which appear in the calculations. For gravity this simply
cannot be done. The resulting quantum field theory is said to be
unrenormalisable and is incapable of giving any useful result.

Because quantum gravity is an attempt to combine two different fields of
physics, there are two distinct groups of physicists involved. These two groups
form a different interpretation of the failure of the direct attack. The
relativists say that it is because gravity cannot be treated perturbatively. To
try to do so destroys the basic principles on which relativity was founded. It
is, for them, no surprise that this should not work. Particle physicists say
that if a field theory is non-renormalisable then it is because it is
incomplete. The theory must be modified and new fields must be added to cancel
divergences.

Supersymmetry The first significant progress in the problem of quantum
gravity was made by particle physicists. They discovered that a new kind of
symmetry called supersymmetry was very important. particles can be classed into
two types; fermions such as quarks and electrons, and bosons such as photons and
Higgs particles. Supersymmetry allows the two types to intermix. With
supersymmetry we have some hope to unify the matter fields with radiation
fields.

Particle physicists discovered that if the symmetry of space-time is extended
to include supersymmetry, then it is necessary to supplement the metric field of
gravity with other matter fields. Miraculously these fields led to cancellations
of many of the divergences in perturbative quantum gravity. This has to be more
than coincidence. At first it was thought that such theories of Supergravity
might be completely renormalisable. After many long calculations this hope
faded.

A funny thing about supergravity was that it works best in ten dimensional
space-time. This inspired the revival of an old theory called Kaluza-Klein
theory, which suggests that space-time has more dimensions than the four obvious
ones. The extra dimensions are not apparent because they are curled up into a
small sphere with a circumference as small as the Planck length. This theory
provides a means to unify the gauge symmetry of general relativity with the
internal gauge symmetries of particle physics.

The next big step taken by particle physicists came along shortly after.
Green and Schwarz realised that a theory which had originally been studied as a
theory of the strong nuclear force was actually more interesting as a theory of
gravity. This was the beginning of string theory. Combining string theory and
supergravity to form superstring theory quickly led to some remarkable
discoveries. A small set of string theories in ten dimensions were perfectly
renormalisable. This was exactly what they were looking for.

It seemed once again that the solution was near at hand, but nature does not
give up its secrets so easily. The problem now was that there is a huge number
of ways to apply Kaluza-Klein theory to the superstring theories. Hence there
seem to be a huge number of possible unified theories of physics. The
perturbative formulation of string theory makes it impossible to determine the
correct way.

Recently there has been renewed hope for string theory from the discovery
that different string theories are connected. They may all be parts of one
unique theory after all.

Canonical Quantum Gravity While particle physicists were making a lot of
noise about superstring theory, relativists have been quietly trying to do
things differently. Many of them take the view that to do quantum gravity
properly you must respect its diffeomorphism symmetry. The Wheeler-DeWitt
equation together with a Hamiltonian constraint equation, describe the way in
which the quantum state vector should evolve according to this canonical
approach.

For a long time there seemed little hope of finding any solutions to the
Wheeler-DeWitt equation. Then in 1986 Ashtekar found a way to reformulate
Einstein's equations of gravity in terms of new variables. Soon afterwards a way
was discovered to find solutions to the equations. This is now known as the loop
representation of quantum gravity. Mathematicians were surprised to learn that
knot theory was an important part of the concept.

The results from the canonical approach seem very different from those of
string theory. There is no need for higher dimensions or extra fields to cancel
divergences. Relativists point to the fact that a number of field theories which
appear to be unrenormalisable have now been quantised exactly. There is no need
to insist on a renormalisable theory of quantum gravity. On the other hand, the
canonical approach still has some technical problems to resolve. It could yet
turn out that the theory can only be made fully consistent by including
supersymmetry.

As well as their differences, the two approaches have some striking
similarities. In both cases they are trying to be understood in terms of
symmetries based on loop like structures. It seems quite plausible that they are
both aspects of one underlying theory. Other mathematical fields are common
features of both, such as knot theory and topology. Indeed there is now a
successful formulation of quantum gravity in three dimensional space-time which
can be regarded as either a loop representation or a string theory. A number of
physicists such as Lee Smolin are looking for a more general common theory
uniting the two approaches.

Black Hole Thermodynamics Although there is no direct empirical input into
quantum gravity, physicists hope to accomplish unification by working on the
requirement that there must exist a mathematically self consistent theory which
accounts for both general relativity and quantum mechanics as they are
separately confirmed experimentally. It is important to stress the point that no
complete theory satisfying this requirement has yet been found. If just one
theory could be constructed then it would have a good chance of being correct.

Because of the stringent constraints that self consistency enforces, it is
possible to construct thought experiments which provide strong hints about the
properties a theory of quantum gravity has to have. There are two physical
regimes in which quantum gravity is likely to have significant effects. In the
conditions which existed during the first Planck unit of time in our universe,
matter was so dense and hot that unification of gravity and other forces would
have been realised. Likewise, a small black hole who's mass corresponds to the
Planck unit of mass provides a thought laboratory for quantum gravity.

Black holes have the property that the surface area of their event horizons
must always increase. This is suggestively similar to the law that entropy must
increase, and it led Bekenstein to conjecture that the area of the event horizon
of a black hole is in fact proportional to its entropy. If this is the case then
a black hole would have to have a temperature and obey the laws of
thermodynamics. In the 1970's Stephen Hawking investigated the effects of
quantum mechanics near a black hole using semi-classical approximations to
quantum gravity. He discovered the unexpected result that black holes do emit
thermal radiation in a way consistent with the entropy law of Bekenstein.

This forces us to conclude that black holes can emit particles and eventually
evaporate. For astronomical sized black holes the temperature of the radiation
is minuscule and certainly beyond detection, but for small black holes the
temperature increases until they explode in one final blast. Hawking realised
that this creates a difficult paradox which would surely tell us a great deal
about the nature of quantum gravity if we could understand it.

The entropy of a system can be related to the amount of information required
to describe it. When objects are thrown into a black hole the information they
contain is hidden from outside view because no message can return from inside.
Now if the black hole evaporates, this information will be lost in contradiction
to the laws of thermodynamics. This is known as the black hole information loss
paradox.

A number of ways on which this paradox could be resolved have been proposed.
The main ones are,

The lost information escapes to another universe The final stage of black
hole evaporation halts leaving a remnant particle which holds the information.
There are strict limits on the amount of information held within any region of
space to ensure that the information which enters a black hole cannot exceed the
amount represented by its entropy. Something else happens which is so strange we
can't bring ourselves to think of it. The first solution would imply a breakdown
of quantum coherence. We would have to completely change the laws of quantum
mechanics to cope with this situation. The second case is not quite so bad but
it does seem to imply that small black holes must have an infinite number of
quantum numbers which would mean their rate of production during the big bang
would have been divergent. It might be possible to find a way round this but
anyway, it is an ugly solution!

Assuming that I have not missed something out, which is a big assumption, we
must conclude that the amount of entropy which can be held within a region of
space is limited by the area of a surface surrounding it. This is certainly
counterintuitive because you would imagine that you could write information on
bits of paper and the amount you could cram in would be limited by the volume
only. This is false because any attempt to do that would eventually cause a
black hole to form. Note that this rule does not force us to conclude that the
universe must be finite because there is a hidden assumption that the region of
space is static which I did not mention.

If the amount of information is limited then the number of physical degrees
of freedom in a field theory of quantum gravity must also be limited. Inspired
by this observation, Gerard 't Hooft, Leonard Susskind and others have proposed
that the laws of physics should be described in terms of a discrete field theory
defined on a space-time surface rather than throughout space-time. They liken
the way this might work to that of a hologram which holds a three dimensional
image within its two dimensional surface.

Rather than being rejected as a crazy idea, this theory has been recognised
by many other physicists as being consistent with other ideas in quantum
gravity.

Quantised Space-time Although there has been considerable progress on the
problem of quantising gravity, it seems likely that it will not be possible to
complete the solution without some fundamental change in the way we think about
space-time. All the approaches I have described suggest that the Planck units of
length and time define a minimum scale of measurement. Indeed the same
conclusion can be reached using fairly general arguments based on the Heisenberg
uncertainty principle applied to the metric field of gravity.

One possibility would be that space-time is some kind of lattice structure at
small scales. A regular cubic lattice structure is generally regarded as an
unacceptable alternative because it destroys space-time symmetry. A random
lattice is more plausible. Numerical studies of statistical randomly
triangulated surfaces are quite encouraging. The Regge calculus describes such a
discretisation of gravity and is akin to topological lattice quantum field
theories as models of quantum gravity in three dimensions.

As far back as 1947, Synder attempted to quantise space-time by treating
space-time co-ordinates as non-commutating operators. The original formulation
was unsuccessful but recent work on quantum groups have initiated a revival of
this approach. This approach also leads to a discrete interpretation of
space-time. Another related topic is non-commutative geometry in which
space-time itself is regarded as secondary to the algebra of fields which can be
generalised to have non-commuting products.

Still this seems to be not quite radical enough to account for quantum
gravity. Some physicists believe that we must modify our views sufficiently to
allow for dynamical changes in the number of space-time dimensions.

To face the quantum gravity challenge we need new insights and new principles
like those which guided Einstein to the correct theory of gravity.